DLMF:15.2.E4 (Q4981)

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DLMF:15.2.E4
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    Statements

    Defining formula
    F ⁡ ( - m , b ; c ; z ) = ∑ n = 0 m ( - m ) n ⁢ ( b ) n ( c ) n ⁢ n ! ⁢ z n = ∑ n = 0 m ( - 1 ) n ⁢ ( m n ) ⁢ ( b ) n ( c ) n ⁢ z n . Gauss-hypergeometric-F 𝑚 𝑏 𝑐 𝑧 superscript subscript 𝑛 0 𝑚 Pochhammer 𝑚 𝑛 Pochhammer 𝑏 𝑛 Pochhammer 𝑐 𝑛 𝑛 superscript 𝑧 𝑛 superscript subscript 𝑛 0 𝑚 superscript 1 𝑛 binomial 𝑚 𝑛 Pochhammer 𝑏 𝑛 Pochhammer 𝑐 𝑛 superscript 𝑧 𝑛 {\displaystyle{\displaystyle F\left(-m,b;c;z\right)=\sum_{n=0}^{m}\frac{{\left% (-m\right)_{n}}{\left(b\right)_{n}}}{{\left(c\right)_{n}}{n!}}z^{n}=\sum_{n=0}% ^{m}(-1)^{n}\genfrac{(}{)}{0.0pt}{}{m}{n}\frac{{\left(b\right)_{n}}}{{\left(c% \right)_{n}}}z^{n}.}}
    0 references
    DLMF id
    DLMF:15.2.E4
    0 references
    Symbols used
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    Defining formula
    F ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}
    xml-id
    C15.S2.E1.m2aadec
    0 references
     
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